We have developed a technique to systematically discover and study high-redshift supernovae that can be used to measure the cosmological parameters. We report here results based on the initial seven of more than 28 supernovae discovered to date in the high-redshift supernova search of the Supernova Cosmology Project. We find an observational dispersion in peak magnitudes of σMB = 0.27; this dispersion narrows to σMB.corr = 0.19 after "correcting" the magnitudes using the light-curve "widthluminosity" relation found for nearby (z ≤ 0.1) Type la supernovae from the Calán/Tololo survey (Hamuy et al). Comparing light-curve width-corrected magnitudes as a function of redshift of our distant (z = 0.35-0.46) supernovae to those of nearby Type la supernovae yields a global measurement of the mass density, ΩM = 0.88-0.60+0.69 for a Λ = 0 cosmology. For a spatially flat universe (i.e., ΩM + ΩΛ = 1), we find ΩM = 0.94-0.28+0.34 or equivalently, a measurement of the cosmological constant, ΩΛ = 0.06-0.34+0.28 (<0.51 at the 95% confidence level). For the more general Friedmann-Lemaître cosmologies with independent ΩM and ΩΛ, the results are presented as a confidence region on the ΩM-ΩΛ plane. This region does not correspond to a unique value of the deceleration parameter q0. We present analyses and checks for statistical and systematic errors and also show that our results do not depend on the specifics of the width-luminosity correction. The results for ΩΛ-versus-ΩM are inconsistent with Λ-dominated, low-density, flat cosmologies that have been proposed to reconcile the ages of globular cluster stars with higher Hubble constant values.